www.biomathforum.org/biomath/index.php/biomath ORIGINAL ARTICLE On the Hausdorff distance between the shifted Heaviside step function and the transmuted Stannard growth function Anton Iliev ∗†, Nikolay Kyurkchiev †, Svetoslav Markov † ∗Faculty of Mathematics and Informatics Paisii Hilendarski University of Plovdiv, Bulgaria aii@uni-plovdiv.bg †Institute of Mathematics and Informatics Bulgarian Academy of Sciences, Sofia, Bulgaria nkyurk@math.bas.bg, smarkov@bio.bas.bg Received: 4 July 2016, accepted: 4 September 2016, published: 14 September 2016 Dedicated to Professor Roumen Anguelov on the occasion of his 60th anniversary Abstract—In this paper we study the one-sided Hausdorff distance between the shifted Heaviside step–function and the transmuted Stannard growth function. Precise upper and lower bounds for the Hausdorff distance have been obtained. We present a software module (intellectual property) within the programming environment CAS Mathematica for the analysis of the growth curves. Numerical examples, illustrating our results are given, too. Keywords-Transmuted Stannard growth function; Heaviside step function; Hausdorff distance; Upper and lower bounds. I. INTRODUCTION AND PRELIMINARIES The Stannard function finds numerous applica- tions in many scientific fields, including population dynamics, bacterial growth, population ecology, plant biology, chemistry, agriculture, demography, financial mathematics, statistics and fuzzy set the- ory [1]–[5]. Definition 1. For γ ∈ R define the shifted Heavi- side step function as [12]: hγ(t) =   0, if t < 0, [0, 1], if t = γ, 1, if t > γ. (1) Definition 2. Define the shifted Stannard growth function S(t) as [1]–[5]: S(t) = 1( 1 + e −(β+k(t−γ)) m )m , (2) where β, k and m ∈ R are the growth parameters. We note that the slope of (2) at t = γ is equal to: ke− β m( 1+e− β m )m+1 . Definition 3. A random variable T is said to have a transmuted distribution if its cumulative distribution function (cdf) is given by [6], [7]: G1(t) = (1 + λ)F1(t) −λF21 (t), |λ| ≤ 1, (3) where F1(t) is the cdf of the base distribution. Citation: Anton Iliev, Nikolay Kyurkchiev, Svetoslav Markov, On the Hausdorff distance between the shifted Heaviside step function and the transmuted Stannard growth function, Biomath 5 (2016), 1609041, http://dx.doi.org/10.11145/j.biomath.2016.09.041 Page 1 of 6 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2016.09.041 A. Iliev et al., On the Hausdorff distance between the shifted Heaviside ... ”λ - transmuting” of (cdf) is a familiar technique from the field of probability distributions with application to insurance mathematics. Definition 4. The Hausdorff distance ρ(f,g) be- tween two interval functions f,g on Ω ⊆ R, is the distance between their completed graphs F(f) and F(g) considered as closed subsets of Ω × R [8], [9], [12]. More precisely, we have ρ(f,g) = max{ sup A∈F(f) inf B∈F(g) ||A−B||, (4) sup B∈F(g) inf A∈F(f) ||A−B||}, wherein ||.|| is any norm in R2, e. g. the maximum norm ||(t,x)|| = max{|t|, |x|}; hence the distance between the points A = (tA,xA), B = (tB,xB) in R2 is ||A−B|| = max(|tA − tB|, |xA −xB|). Sigmoidal growth curves typically have three parts (phases, time intervals): lag, log and station- ary parts. It is a challenging question to charac- terize mathematically these phases. The lag time (interval) is practically important in many medical and biotechnological applications as this time is responsible for the acceleration or inhibition of the process and the possibility of controlling the lag time depends on the understanding of the hidden mechanisms of the corresponding process [10], [11]. Usually the lag time is defined by means of the uniform distance between the sigmoidal function and the induced cut function. We propose a new definition for the lag time by means of the Haus- dorff distance between the sigmoidal function and the induced step function. In this work we prove estimates for the one– sided Hausdorff approximation of the shifted Heaviside step–function by transmuted Stannard growth function. Let us point out that the Hausdorff distance is a natural measuring criteria for the approximation of bounded discontinuous functions [12], [13]. Fig. 1. Approximation of the shifted Heaviside step function by transmuted Stannard growth function for the following data: k = 16, m = 0.52, β = 0.01, tr = 5; Hausdorff distance d = 0.0801797. Fig. 2. Approximation of the shifted Heaviside step function by transmuted Stannard growth function for the following data: k = 26, m = 2.1, β = 1, tr = 5; Hausdorff distance d = 0.112237. II. MAIN RESULTS For γ,β,m ∈ R consider the following trans- muted Stannard function S∗(t) = 1 + λ( 1 + e −(β+k(t−γ)) m )m− (5) λ( 1 + e −(β+k(t−γ)) m )2m , |λ| ≤ 1. Biomath 5 (2016), 1609041, http://dx.doi.org/10.11145/j.biomath.2016.09.041 Page 2 of 6 http://dx.doi.org/10.11145/j.biomath.2016.09.041 A. Iliev et al., On the Hausdorff distance between the shifted Heaviside ... Function S∗(t) from (5) satisfies: S∗(γ) = 1 + λ( 1 + e− β m )m − λ( 1 + e− β m )2m = 12 (6) hence λ = 0.5(1 + z)2m − (1 + z)m (1 + z)m − 1 ; z = e− β m . (7) We study the Hausdorff approximation d of the Heaviside step function hγ(t) by the trans- muted Stannard function (5)–(7) and look for an expression for the error of the best one–sided approximation. Let A = (1 + λ) ( 1 + e− β m )−m −λ ( 1 + e− β m )−2m B = 1 − 2e− β m ( 1 + e− β m )−1−2m kλ + e− β m ( 1 + e− β m )−1−m k(1 + λ), k ∈ R. (8) The following Theorem gives upper and lower bounds for d. Theorem 2.1 For the Hausdorff distance d between the function hγ(t) and the transmuted Stannard function (5)–(7) the following inequal- ities hold for |λ| ≤ 1 and B > 4: dl = A 2B < d < A ln(2B) 2B = dr. (9) Proof. We need to express d in terms of k, β and m. The Hausdorff distance d satisfies the relation F(d) := S∗(γ −d) = 1 + λ( 1 + e− β−kd m )m− (10) λ( 1 + e− β−kd m )2m −d = 0. Consider the function G(d) = A−Bd. Fig. 3. The functions F(d) and G(d) for k = 16, m = 0.52, β = 0.01, tr = 5. By means of Taylor expansion we obtain G(d) −F(d) = O(d2). Hence G(d) approximates F(d) with d → 0 as O(d2) (see Fig. 3). Further, for |λ| ≤ 1 and B > 4 we have G(dl) = A 2 > 0, G(dr) = A (1 − 0.5 ln(2B)) < 0. This completes the proof of the theorem. Some computational examples using relations (9) are presented in Table 1. The last column of Table 1 contains the values of d computed by solving the nonlinear equation (10). TABLE I BOUNDS FOR d COMPUTED BY EQUATION (9) FOR VARIOUS β, k, m. Biomath 5 (2016), 1609041, http://dx.doi.org/10.11145/j.biomath.2016.09.041 Page 3 of 6 http://dx.doi.org/10.11145/j.biomath.2016.09.041 A. Iliev et al., On the Hausdorff distance between the shifted Heaviside ... Fig. 4. A simple module implemented in CAS Mathematica for the computation and visualization of the Hausdorff distance between the Heaviside step function and the transmuted Stannard growth function. Biomath 5 (2016), 1609041, http://dx.doi.org/10.11145/j.biomath.2016.09.041 Page 4 of 6 http://dx.doi.org/10.11145/j.biomath.2016.09.041 A. Iliev et al., On the Hausdorff distance between the shifted Heaviside ... III. CONCLUSION REMARKS New estimates for the Hausdorff distance be- tween an interval Heviside step function and its best approximating Stannard function are obtained. On Fig. 1 and Fig. 2 appropriate illustrations of some approximations of the shifted Heaviside step function by transmuted Stannard growth function are given. We propose a software module within the pro- gramming environment CAS Mathematica for the analysis of the considered growth curves (see Fig. 4). The module offers the following possibilities: i) generation of the shifted Stannard curve under user-defined values for k,m,β; ii) automatic check of the condition |λ| ≤ 1 that guarantees the existence of sigmoidality of the transmuted Stannard curve; iii) software tools for animation and visualiza- tion. The Hausdorff approximation of the interval step function by the logistic and other sigmoidal functions is discussed from various approximation, computational and modelling aspects in [14]–[27]. ACKNOWLEDGMENTS The authors would like to thank the anony- mous reviewers for their helpful comments that contributed to improving the final version of the presented paper. REFERENCES [1] C. Stannard, A. Williams, P. Gibbs, Tempera- ture/growth relationship for psychotropic food– spoilage bacteria, Food Microbiol. 2 (1985) 115–122. [2] A. Khamis, Z. Ismail, Kh. Haron, A. Mohammed, Nonlinear growth models for modeling oil palm yield growth, Journal of Mathematics and Statistics 1(3) (2005) 225–233, doi:10.3844/jmssp.2005.225.232 [3] M. Zwietering, I. Jongenburger, F. Rombouts, K. Riet, Modeling of the Bacterial Growth Curve, Appl. and Environmental Microbiology 56(6) (1990) 1875–1881. [4] M. Panik, Growth curve modeling: theory and appli- cations, Wiley (2014), ISBN: 978-1-118-76404-6, doi:10.1002/9781118763971 [5] N. Kyurkchiev, A. Iliev, On some growth curve mod- eling: approximation theory and applications, Int. J. of Trends in Research and Development, 3(3) (2016) 466–471. [6] W. Shaw, I. Buckley, The alchemy of probability distri- butions: beyond Gram-Charlier expansion and a skew- kurtotic-normal distribution from rank transmutation map, Research report (2007). [7] R. C. Gupta, O. Acman, S. Lvin, A sudy of log–logistic model in survival analysis, Biometrica Journal, 41(4) (1999) 431–443, doi:10.1002/(SICI)1521-4036(199907) 41:4<431::AID-BIMJ431>3.0.CO;2-U [8] F. Hausdorff, Set Theory (2 ed.), Chelsea Publ., New York (1962 [1957]) (Republished by AMS-Chelsea 2005), ISBN: 978–0–821–83835–8. [9] B. Sendov, Hausdorff Approximations, Kluwer, Boston (1990) doi:10.1007/978-94-009-0673-0 [10] S. Shoffner, S. Schnell, Estimation of the lag time in a subsequent monomer addition model for fibril elongation, bioRxiv The preprint server for biology (2015) 1–8, doi:10.1101/034900 [11] P. Arosio, T. P. J. Knowles, S. Linse, On the lag phase in amyloid fibril formation, Physical Chemistry Chemical Physics 17 (2015) 7606–7618, doi:10.1039/C4CP05563B [12] R. Anguelov, S. Markov, Hausdorff Continuous Inter- val Functions and Approximations, In: M. Nehmeier et al. (Eds), Scientific Computing, Computer Arithmetic, and Validated Numerics, 16th International Sympo- sium, Springer SCAN 2014, LNCS 9553 (2016) 3–13, doi:10.1007/978-3-319-31769-4 [13] N. Kyurkchiev, A. Andreev, Approximation and an- tenna and filter synthesis: Some moduli in program- ming environment Mathematica, LAP LAMBERT Aca- demic Publishing, Saarbrucken (2014), ISBN 978-3- 659-53322-8. [14] D. Costarelli, R. Spigler, Approximation results for neural network operators activated by sigmoidal func- tions, Neural Networks 44 (2013) 101–106, doi:10.1016/j.neunet.2013.03.015 [15] N. Kyurkchiev, On the Approximation of the step func- tion by some cumulative distribution functions, Compt. rend. Acad. bulg. Sci. 68(12) (2015) 1475–1482. [16] N. Kyurkchiev, S. Markov, On the Hausdorff distance between the Heaviside step function and Verhulst lo- gistic function, J. Math. Chem. 54(1) (2016) 109–119, doi:10.1007/S10910-015-0552-0 [17] N. Kyurkchiev, S. Markov, Sigmoidal functions: some computational and modelling aspects, Biomath Com- munications 1(2) (2014) 30–48, doi:10.11145/j.bmc.2015.03.081 [18] A. Iliev, N. Kyurkchiev, S. Markov, On the Approxi- mation of the Cut and Step Functions by Logistic and Gompertz Functions, BIOMATH 4(2) (2015) 1510101, doi:10.11145/j.biomath.2015.10.101 [19] A. Iliev, N. Kyurkchiev, S. Markov, On the Approxi- mation of the step function by some sigmoid functions, Biomath 5 (2016), 1609041, http://dx.doi.org/10.11145/j.biomath.2016.09.041 Page 5 of 6 http://dx.doi.org/10.11145/j.biomath.2016.09.041 A. Iliev et al., On the Hausdorff distance between the shifted Heaviside ... Mathematics and Computers in Simulation (2015), doi:10.1016/j.matcom.2015.11.005 [20] N. Kyurkchiev, S. Markov, On the approximation of the generalized cut function of degree p+1 by smooth sigmoid functions, Serdica J. Computing 9(1) (2015) 101–112. [21] N. Kyurkchiev, S. Markov, Sigmoid functions: Some Approximation and Modelling Aspects, LAP LAM- BERT Academic Publishing, Saarbrucken (2015), ISBN 978-3-659-76045-7. [22] V. Kyurkchiev, N. Kyurkchiev, On the Approximation of the Step function by Raised-Cosine and Laplace Cumulative Distribution Functions, European Interna- tional Journal of Science and Technology 4(9) (2015) 75–84. [23] N. Kyurkchiev, S. Markov, A. Iliev, A note on the Schnute growth model, Int. J. of Engineering Research and Development 12(6) (2016) 47–54. [24] N. Kyurkchiev, A. Iliev, A note on some growth curves arising from Box-Cox transformation, Int. J. of Engi- neering Works 3(6) (2016) 47–51. [25] N. Kyurkchiev, A note on the new geometric repre- sentation for the parameters in the fibril elongation process, Compt. rend. Acad. bulg. Sci. 69(8) (2016) 963–972. [26] D. Costarelli, G. Vinti, Pointwise and uniform approxi- mation by multivariate neural network operators of the max-product type, Neural Networks (2016), doi:10.1016/j.neunet.2016.06.002 [27] Iliev, A., N. Kyurkchiev, S. Markov, Approximation of the cut function by Stannard and Richards sigmoid functions, IJPAM 109(1) 2016 119–128. Biomath 5 (2016), 1609041, http://dx.doi.org/10.11145/j.biomath.2016.09.041 Page 6 of 6 http://dx.doi.org/10.11145/j.biomath.2016.09.041 Introduction and preliminaries Main Results Conclusion remarks References